On Solving Fractional Higher-Order Equations via Artificial Neural Networks

Recently, different structures of artificial neural networks (shortly ANNs) have been proposed for the modeling and simulation of many real-world complex phenomena. The current research is devoted to the numerical study of an ordinary linear fractional-order integro-differential equation of Volterra type. By substituting the unknown function with a suitable three-layered feed-forward neural architecture, this initial value fractional problem is converted approximately to a system of nonlinear minimization equations. Due to the complexity of the achieved problem, the back-propagation algorithm is employed by making small adjustments in the learning process. In other words, an iterative optimization algorithm based on the gradient descent method is constructed to approximate the solution of the origin fractional problem. Moreover, some examples consist of computer simulations are provided to demonstrate the accuracy and ability of the indicated iterative technique. The obtained numerical results show the efficiency and capability of the ANNs approach in comparison with traditional methods.